Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Trigonometric functions wikipedia, lookup

Pythagorean theorem wikipedia, lookup

Transcript
```Honors Geometry Section 5.2
Areas of Triangles and
WHY?
RHL
E
F
You can see that the area of parallelogram
ABCD is equal to the area of rectangle EBCF.
For a parallelogram with base b
and height h, the area is given by
the formula:
A parallelogram = ______
b h
Note that the height is the length of the
segment perpendicular to the base from a
point on the opposite side which is called
the altitude of the parallelogram.
2s
s 3 4 3
s
+
A  15  4 3  60 3 u
2
A  10  6  60
8  x  60
x  60 / 8  7.5 u
Any triangle is half of a parallelogram. For a
triangle with base b and height h, the area
is given by the formula:
1 bh
A triangle = ________
2
The height is the length of the ____________
altitude to
the base
Example: Find the area of to the nearest 1000th.
AC
sin 25 
10
AC  4.226
BC
cos 25 
10
BC  9.063
A  .5  4.226  9.063  19.150 u
2
Example: A triangle has an area of 56
and a base of 10. Find its height.
A  1 bh
2
56  1 10h
2
h  11.2
A  .5(10)(3)  15 u
2
Trigonometry and the
Area of a Triangle
Using your knowledge of trigonometry, express h in terms of
sinC.
h
sin C 
b
b sin C  h
1
Substituting this into the formula A  bh , and using a as the
2
base we get
1
A  ab sin C
2
We have just discovered that the
area of a triangle can be expressed
using the lengths of two sides and
the sine of the included angle.
Example: Use what you have learned above to find
the area of parallelogram ABCD to the nearest 1000th.
A// gram  2( Atriangle )
A// gram
1

 2 15  25  sin 50 
2

A// gram  15  25  sin 50
A// gram  287.267cm
2
An altitude of a trapezoid is a
segment perpendicular to the two
bases with an endpoint in each of
the bases.
The length of an altitude will be the height
of the trapezoid.
1 b2 h
2
1 b1h
2
1 h(b2  b1 )
2
1 b2h  1 b1h
2
2
For a trapezoid with bases b1 and b2 and
height h, the area of a trapezoid is given by
the formula:
1
Atrap. 
2
hb1  b2 
Recall that the diagonals of both
rhombuses and kites are
perpendicular.
1
1
Akite  1

BC  AE 
2

BC DE 
2
E

BC  AE   1 BC DE 
2
2
Akite  1

BC  AE  DE 
2
1
Akite  Ar hombus 
d1d 2
2
```